Hi Andre,
Excluding Hakone, all your destinations are easily accessible with the JRPass. My recommendation would be to buy a JRPass in combination with a Hakone FreePass. It favours you quite well as the Hakone pass is for 2-3 days, which fits into your schedule.
To get to Zao Onsen from Yamagata, you'll need to purchase a bus ticket as well. A round trip will cost 2000¥, which is about $14. To do this, you'll have to alight at Yamagata Station, and head to the Yamako Bus Information Center to purchase a ticket. If you type it into your browser information and directions will come up.
More info on this can be found here: https://travel.yamakobus.jp/en/timetable/routebus-yamagata-zaoline.html
As for train schedules, your best bet is using Google maps. If you put your starting point, destination, date, and time, it will tell you how to get there. Unfortunately the schedules provided by the JR companies aren't particularly accessible.
Hope this helps! If you have any further questions, please don't hesitate to ask.